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EURASIP Journal on Advances in Signal ProcessingVolume 2008, Article ID 289184, 11 pages doi:10.1155/2008/289184 Research Article Multirate Formulation for Mismatch Sensitivity Analysis

EURASIP Journal on Advances in Signal Processing

Volume 2008, Article ID 289184, 11 pages

doi:10.1155/2008/289184

Research Article

Multirate Formulation for Mismatch Sensitivity

Analysis of Analog-to-Digital Converters That Utilize

Anton Blad, H˚ akan Johansson, and Per L ¨owenborg

Division of Electronics Systems, Department of Electrical Engineering, Link¨oping University, Sweden

Correspondence should be addressed to Anton Blad,antonb@isy.liu.se

Received 1 June 2007; Accepted 21 October 2007

Recommended by Boris Murmann

A general formulation based on multirate filterbank theory for analog-to-digital converters using parallel sigmadelta modulators

in conjunction with modulation sequences is presented The time-interleaved modulators (TIMs), Hadamard modulators (HMs), and frequency-band decomposition modulators (FBDMs) can be viewed as special cases of the proposed description The useful-ness of the formulation stems from its ability to analyze a system’s sensitivity to aliasing due to channel mismatch and modulation sequence level errors Both Nyquist-rate and oversampled systems are considered, and it is shown how the matching requirements between channels can be reduced for oversampled systems The new formulation is useful also for the derivation of new modula-tion schemes, and an example is given of how it can be used in this context

Copyright © 2008 Anton Blad et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Traditionally, analog-to-digital converters (ADCs) and

digital-to-analog converters (DACs) based on

ΣΔ-modula-tion have been used primarily for low bandwidth and

high-resolution applications such as audio application The

re-quirements make the architecture perfectly suited for this

purpose However, in later years, advancements in VLSI

tech-nology have allowed greatly increased clock frequencies, and

ΣΔ-ADCs with bandwidths of tens of MHz have been

re-ported [1, 2] This makes it possible to use ΣΔ-ADCs in

a wider context, for example, in wireless communications

One of the most attractive features ofΣΔ-ADCs is their

re-laxed requirements on the analog circuitry, which is

espe-cially important in wireless communications where

integra-tion in analog-hostile deep submicron CMOS is favorable

However, the high-operating frequencies used for the

realiza-tion of such wideband converters result in devices with high

analog power consumption

One way to reduce the operating frequency is to use

sev-eral modulators in parallel, where a part of the input signal is

converted in each channel Several flavors of suchΣΔ-ADCs

have been proposed, and these can essentially be divided into

four categories: time-interleaved modulators (TIMs) [3,4], Hadamard modulators (HMs) [4 8], frequency-band de-composed modulators (FBDMs) [4, 9, 10] and multirate modulators based on block-digital filtering [11–14] In the TIM, samples are interleaved in time between the channels Each modulator is running at the input sampling rate, with its input grounded between consecutive samples This is a simple scheme, but as interleaving causes aliasing of the spec-trum, the channels have to be carefully matched in order to cancel aliasing in the deinterleaving at the output In an HM, the signal is modulated by a sequence constructed from the rows of a Hadamard matrix One advantage over the TIM is

an inherent coding gain, which increases the dynamic range

of the ADC [4], whereas a disadvantage is that the number

of channels is restricted to a number for which there exists

a known Hadamard matrix Another advantage, as will be shown in this paper, is the reduced sensitivity to mismatches

in the analog circuitry The third category of parallel mod-ulators is the FBDM, in which the signal is decomposed in frequency rather than time This scheme is insensitive to ana-log mismatches, but has increased hardware complexity be-cause it requires the use of bandpass modulators The idea of the multirate modulators is different, based on a polyphase

decomposition of the integrator in one channel Thus the

ar-chitecture is not directly comparable to the systems described

in this paper

The parallel systems have been analyzed both in the

time-domain and the frequency-time-domain [3, 4,6 8,12,15–17],

and in [18] an attempt was made to formulate a general

model covering the TIM, HM, and FBDM systems The

for-mulation in this paper is slightly different from the one in

[18] due to differences in the usage of causal and noncausal

delays The overall ADC was formulated in terms of

circu-lant and pseudocircucircu-lant matrices, and the formulation is

derived from multirate filter bank theory The formulation

is refined in this paper, and extended with a more

compre-hensive sensitivity analysis Using the formulation, the

be-havior of a practical ADC with channel gain and modulation

sequence level mismatches present can be analyzed, and it is

apparent why some schemes are sensitive to mismatches

be-tween channels whereas others are not Also, it is found that

some schemes (in particular the HM systems) suffer from

sensitivity in a limited set of channels such that “full

calibra-tion” between the channels is not needed Whereas the new

formulation is in fact not constrained to parallelΣΔ-ADCs

but applicable to general parallel systems that use

modula-tion sequences, it is described in that context in this paper as

this application is considered to be particularly interesting

Further, the usefulness of the new formulation is not only

limited to the analysis of existing schemes, but also for the

derivation of new ones, which is demonstrated in the paper

The organization of the paper is as follows InSection 2,

the multirate formulation of a parallel system is derived, and

the signal input-to-output relation of the system is analyzed

Conditions for the system to be free from nonlinear

distor-tion (i.e., free from aliasing) are stated InSection 3, the

sen-sitivity to channel mismatches for a system is analyzed in the

context of the multirate formulation InSection 4, the

for-mulation is used to analyze the behavior of some

representa-tive systems, and also the derivation of a new scheme that is

insensitive to some mismatches is presented InSection 5, the

quantization noise properties of a parallel system is analyzed

Finally,Section 6concludes the paper

We consider the scheme depicted inFigure 1 In this scheme,

the input signalx(n) is first divided into N channels In each

channelk, k = 0, 1, , N −1, the signal is first modulated

by theM-periodic sequence a k(n) = a k(n + M) The

result-ing sequence is then fed into aΣΔ-modulator ΣΔk, followed

by a digital filterG k(z) The output of the filter is modulated

by theM-periodic sequence b k(n) = b k(n + M) which

pro-duces the channel output sequencey k(n) Finally, the overall

output sequence y(n) is obtained by summing all channel

output sequences TheΣΔ-modulator in each channel works

in the same way as an ordinaryΣΔ-modulator By increasing

the channel oversampling, and reducing the passband width

of the channel filter accordingly, most of the shaped noise

is removed, and the resolution is increased By using

sev-eral channels in parallel, wider signal bands can be handled

without increasing the input sampling rate to unreasonable

×

×

×

.

x(n)

a0 (n)

a1 (n)

a N−1(n)

ΣΔ 0

ΣΔ 1

ΣΔN−1

G0 (z)

G1 (z)

G N−1(z) ×

×

×

b0 (n)

b1 (n)

b N−1(n)

y N−1(n)

y1 (n)

y0 (n)

+ y(n)

Figure 1: ADC system using parallelΣΔ-modulators and modula-tion sequences

values In other words, instead of using one singleΣΔ-ADC with a very high input sampling rate, a number ofΣΔ-ADCs

in parallel provide essentially the same resolution but with a reasonable input sampling rate

The overall outputy(n) is determined by the input x(n),

the signal transfer function of the system, and the quanti-zation noise generated in theΣΔ-modulators Using a linear model for analysis, the signal input-to-output relation and noise input-to-output relation can be analyzed separately The signal transfer function from x(n) to y(n) should be

equal to (or at least approximate) a delay in the signal fre-quency band of interest The main problem in practice is that the overall scheme is subject to channel gain, offset, and modulation sequence level mismatches [4, 15,16] This is where the new general formulation becomes very useful as

it gives a relation between the input and output from which one can easily deduce a particular scheme’s sensitivity to mis-match errors The noise contribution, on the other hand, is essentially unaffected by channel mismatches Therefore, the noise analysis can be handled in the traditional way, as in Section 5

From the signal input-to-output point of view, we have the system depicted in Figure 2(a) for channel k Here, each

H k(z) represents a cascade of the corresponding signal

trans-fer function of theΣΔ-modulator and the digital filter G k(z).

To derive a useful input-output relation in thez-domain, we

make use of multirate filter bank theory [19] Asa k(n) and

b k(n) are M-periodic sequences, each multiplication can be

modelled asM branches with constant multiplications and

the samples interleaved between the branches This is shown

in the structure inFigure 2(b), where

a k,n =

⎧

⎨

⎩

a k(0) forn =0,

a k(M −1) forn =1, 2, , M −1,

b k,n = b k(M −1− n) forn =0, 1, , M −1.

(1)

Now, consider the system shown inFigure 3, representing the path fromx q(m) to y k,r(m) inFigure 2(b) Denoting



H (z) = z M −1H (z), (2)

x(n) ×

a k(n)

H k(z) ×

b k(n)

y k(n)

(a) Model of channel

z −1

.

.

z −1

x(n)

↓ M

x0 (m)



x1 (m)



↓ M

x M−1(m)



↓ M

a k,0

a k,1

a k,M−1 ↑ M

↑ M

↑ M +

z

+

.

z

H k(z) z

z

↓ M

↓ M

↓ M

b k,0

b k,1

b k,M−1

y k,0(m)



y k,1(m)



y k,M−1(m)



↑ M

↑ M

↑ M

z −1

+ y k(n)

.

.

+

z −1

(b) Polyphase decomposition of multipliers

z −1

.

z −1 x(n)

↓ M

↓ M

↓ M

x0 (m)



x1 (m)



x M−1(m)



Pk(z)

y k,0(m)



y k,1(m)



y k,M−1(m)



↑ M

↑ M

↑ M

z −1

+ y k(n)

+

z −1

(c) Multirate formulation of a channel

Figure 2: Equivalent signal transfer models of a channel of the parallel system inFigure 1

the transfer function fromx q(m) to y k,r(m) is given by the

first polyphase component in the polyphase decomposition

ofz q Hk(z)z − r, scaled bya k,q b k,r Forp = q − r =0, 1, , M −

1, the polyphase decomposition ofz p Hk(z) can be written

z p Hk(z) = M−1

i =0

z p − i Hk,iz M

and the first polyphase component isHk,p(z), that is, the pth

polyphase component ofHk(z) as specified by the Type 1

polyphase representation in [19] Forp = − M + 1, , −1,

z p Hk(z) =

M−1

i =0

z p − i+M z − M Hk,iz M

(4)

and the first polyphase component isz −1Hk,p+M(z)

Return-ing to the system inFigure 2(b), the transfer functionsP k r,q(z)

fromx q(m) to y k,r(m) can now be written

P r,q k (z) =

⎧

⎨

⎩

b k,r Hk,q − r(z)a k,q forq ≥ r,

b k,r z −1Hk,q − r+M(z)a k,q forq < r. (5)

The relations can be written in matrix form as Pk(z) in

Pk(z) =

⎡

⎢

⎢

⎢

⎢

⎢

A1 A5 A9 · · · A13

A2 A6 A10 · · · A14

A3 A7 A11 · · · A15

. . .

A A A · · · A

⎤

⎥

⎥

⎥

⎥

⎥

where

A1= a k,0 b k,0 Hk,0(z), A2= a k,0 b k,1 z −1Hk,M −1(z),

A3= a k,0 b k,2 z −1Hk,M −2(z), A4= a k,0 b k,M −1z −1Hk,1(z),

A5= a k,1 b k,0 Hk,1(z), A6= a k,1 b k,1 Hk,0(z),

A7= a k,1 b k,2 z −1Hk,M −1(z), A8= a k,1 b k,M −1z −1Hk,2(z),

A9= a k,2 b k,0 Hk,2(z), A10= a k,2 b k,1 Hk,1(z),

A11= a k,2 b k,2 Hk,0(z), A12= a k,2 b k,M −1z −1Hk,3(z),

A13= a k,M −1b k,0 Hk,M −1(z), A14= a k,M −1b k,1 Hk,M −2(z),

A15= a k,M −1b k,2 Hk,M −3(z), A16= a k,M −1b k,M −1Hk,0(z),

(7)

and it is thus obvious that one channel of the system can be represented by the structure inFigure 2(c) In the whole sys-tem (Figure 1) a number of such channels are summed at the output, and the parallel system ofN channels can be

repre-sented by the structure inFigure 4, where the matrix P(z) is

given by

P(z) =

N−1

k =0

For convenience, we write (6) as

x q(m)

a k,q

↑ M z q H k(z) z M−1 z −r ↓ M

b k,r

y k,r(m)

Figure 3: Path fromx q(m) to y k,r(m) in channel k as depicted in

Figure 2(b)

where “·” denotes elementwise multiplication and where



Hk(z) and S kare given by



Hk(z) =

⎡

⎢

⎢

⎢

⎢

⎢

⎣



H k,0(z) Hk,1(z) · · ·  H k,M −1(z)

z −1Hk,M −1(z) Hk,0(z) · · ·  H k,M −2(z)

z −1Hk,M −2(z) z −1Hk,M −1(z) · · ·  H k,M −3(z)

z −1Hk,1(z) z −1Hk,2(z) · · · Hk,0(z)

⎤

⎥

⎥

⎥

⎥

⎥

⎦ (10)

Sk =

⎡

⎢

⎢

⎢

⎢

a k,0 b k,0 a k,1 b k,0 · · · a k,M −1b k,0

a k,0 b k,1 a k,1 b k,1 · · · a k,M −1b k,1

a k,0 b k,2 a k,1 b k,2 · · · a k,M −1b k,2

a k,0 b k,M −1 a k,1 b k,M −1 · · · a k,M −1b k,M −1

⎤

⎥

⎥

⎥

⎥. (11) Equation (11) can equivalently be written as

Sk =bT

where

ak =a k,0 a k,1 · · · a k,M −1

 ,

bk =b k,0 b k,1 · · · b k,M −1

andT stands for transpose Examples of the S k-matrices and

of the ak- and bk-vectors are provided for the TIM system in

(26) and (25) in Example 1inSection 4 Examples are also

provided for the HM and FBDM systems in Examples2and

3, respectively

2.1.1 Alias-free system

With the system represented as above, it is known that it is

alias-free, and thus time-invariant if and only if the matrix

P(z) is pseudocirculant [19] Under this condition, the

out-putz-transform becomes

Y (z) = HA(z)X(z), (14) where

HA(z) = z − M+1

N−1

k =0

M−1

i =0

s0,k i z − i Hk,iz M

=

N−1

k =0

M−1

i =0

s0,k i z − i H k,i



z M ,

(15)

withs0,k idenoting the elements on the first row of Sk This

case corresponds to a Nyquist sampled ADC of which two

z −1

.

z −1 x(n)

↓ M

↓ M

↓ M

P(z)

↑ M

↑ M

↑ M

z −1

+ y(n)

+

z −1

Figure 4: Equivalent representation of the system inFigure 1based

on the equivalences inFigure 2 P(z) is given by (8)

special cases are the TIM ADC [3,12] and HM ADC in [6] These systems are also described in the context of the multi-rate formulation in Examples1and2inSection 4

RegardingHk(z), it is seen in (10) that it is pseudocir-culant for an arbitrary Hk(z) It would thus be sufficient

to make Sk circulant for each channelk in order to make

each Pk(z) pseudocirculant and end up with a

pseudocircu-lant P(z) Unfortunately, the set of circulant real-valued S k

achievable by the construction in (12) is seriously limited,

because the rank of Skis one However, for purposes of er-ror cancellation between channels it is beneficial to group the channels in sets where the matrices within each set sum to a circular matrix The channel set{0, 1, , N −1}is thus par-titioned into the setsC0, , C I −1, where each sum



k ∈ C i

is a circulant matrix It is assumed that the modulators and filters are identical for channels belonging to the same

par-tition, Hk(z) =Hl(z) whenever k, l ∈ C i, and thusHk(z) =



Hl(z) The matrix for partition i is denotedH0,i(z) Sensitiv-ity to channel mismatches are discussed further inSection 3

2.1.2 L-decimated alias-free system

We say that a system is anL-decimated alias-free system if it

is alias-free before decimation by a factor ofL A channel of

such a system is shown inFigure 5(a) Obviously, the deci-mation can be performed before the modulation, as shown

in Figure 5(b), if the index of the modulation sequence is scaled by a factor ofL Considering the equivalent system in

Figure 5(c), it is apparent that the downsampling byL can be

moved to after the scalings byb k,l if the delay elementsz −1

are replaced byL-fold delay elements z − L The system may then be described as inFigure 5(d), where Pk(z) is defined

by (5) However, the outputs are taken from everyLth row of

Pk(z), such that the first output y k,L −1 modM(m) is taken from

rowL, the second output y k,2L −1 modM(m) is taken from row

(2L −1 modM) + 1, and so on It is thus apparent that only

rows gcd(L, M) · i, i =0, 1, 2, , are used.

TheL-decimated system corresponds to an oversampled

ADC The main observation that should be made is that the

x(n) ×

a k(n)

H k(z) ×

b k(n)

↓ L y k(l)

(a) Decimation at output

x(n) ×

a k(n)

H k(z) ×

b k(L1)

↓ L y k(l)

(b) Internal decimation

z −1

.

.

z −1

x(n)

↓ M

x0 (m)



x1 (m)



↓ M

x M−1(m)



↓ M

a k,0

a k,1

a k,M−1 ↑ M

↑ M

↑ M +

z

+

.

z

↓ L

H k(z) z

z

↓ M

↓ M

↓ M

b k,L−1 mod M

b k,2L−1 mod M

b k,ML−1 mod M = b k,M−1

↑ M

↑ M

↑ M

z −1

+

y k(l)

.

.

+

z −1

(c) Polyphase decomposition of input and output

z −1

.

z −1 x(n)

↓ M

↓ M

↓ M

x0 (m)



x1 (m)



x M−1(m)



Pk(z)

y k,L−1 mod M(m)



y k,2L−1 mod M(m)



y k,M−1(m)



↓ L

↓ L

↓ L

↑ M

↑ M

↑ M

z −1

+ y k(l)

+

z −1

(d) Multirate formulation of a channel.y k,L(m) denotes the output

pertaining to theLth row of P k(z)

Figure 5: Channel model ofL-decimated system.

L-decimated system may be described in the same way as

the critically sampled system, but that relaxations may be

allowed on the requirements of the modulation sequences

As only a subset of the rows of P(z) are used, the matrix

needs only to be pseudocirculant on these rows As in the

critically sampled (nonoversampled) case, the channel set

{0, 1, , N −1}is partitioned into setsC0, , C I −1where

the matrix 

k ∈ C iSk is circulant on the rows gcd(L, M) · i,

i =0, 1, 2, , andHk(z) = Hl(z) = H0,i(z) when k, l ∈ C i.

The oversampled Hadamard-modulated system in [7]

be-longs to this category of the formulation, and another

exam-ple of a decimated system is given inExample 4inSection 4

3 SENSITIVITY TO CHANNEL MISMATCHES

In this section, the channel model used for the sensitivity

analysis is explained In the system shown inFigure 6, several

nonidealities resulting from imperfect analog circuits have

been included Difficulties in realizing the exact values of the

analog modulation sequence are modelled by an additive

er-ror termε k(n) The error is assumed to be static, that is, it

depends only on the value ofa k(n), and is therefore a

peri-odic sequence with the same periperi-odicity asa k(n) The

time-varying errorε k(n) may be a major concern when the

mod-ulation sequences contain nontrivial elements, that is,

ele-ments that are not−1, 0, or 1 The trivial elements may be

realized without a multiplier by exchanging, grounding, or

passing through the inputs to the modulator, and are for this

reason particularly attractive on the analog side

A channel-specific gain γ k is included in the

sensitiv-ity analysis, and analog imperfections in the modulator are

modelled as the transfer function ΔH k(z) The modulator

nonidealities including channel gain and modulation

se-quence errors are analyzed separately in the context of the

multirate formulation In practice, there is also a channel o

ff-set δ k which is not suitable for analysis in this context, as

it is signal independent Channel offsets are commented in Section 3.3below

Assume that the ideal system is alias-free, that is, the

ma-trix P(z) = Pk(z) is pseudocirculant Due to analog

cir-cuit errors the transfer function of channelk deviates from

the idealH k(z) to γ k(H k(z) + ΔH k(z)), and Hk(z) is replaced

byHk(z) = γ k(H k(z) + ΔH k(z))z M −1 The transfer matrix for channelk thus becomesPk(z) with elements



P k j,i(z) =

⎧

⎨

⎩

b k, j Hk,i − j(z)a k,i fori ≥ j,

b k, j z −1Hk,i − j+M(z)a k,i fori < j, (17)

whereHk,p(z) are the polyphase components of Hk(z) It is apparent that Pk(z) is pseudocirculant whenever P k(z) is.

Thus a system where all the Skmatrices are circulant is com-pletely insensitive to modulator mismatches

In the general case, unfortunately, all Skare not circulant and Pk(z) = Sk· Hk(z) does not sum up to a

pseudo-circulant matrix as the matricesHk(z) are different between the channels Partitioning the channel set into the sets C i,

as described inSection 2, and matching the modulators of channels belonging to the same partitionC i, that is, defining

γ k = γ landΔH k(z) = ΔH l(z) when k, l ∈ C i, allowsP( z) to

be written



P(z) =

N−1

k =0

Sk· Hk(z) =

I −1



i =0



H0,i(z) ·

k ∈ C

Sk, (18)

and it is apparent that each term in the outer sum is

pseu-docirculant, and thus that also P(z) is Thus the system is

alias-free and non-linear distortion is eliminated

It is assumed that the ideal system is alias-free, that is, P(z) =



Pk(z) is pseudocirculant Due to difficulties in realizing

the analog modulation sequence, the signal is modulated in

channelk by the sequenceak =ak+ε krather than the ideal

sequence ak We consider here different choices of the

mod-ulation sequences

3.2.1 Bilevel sequence for an insensitive channel

Assume that an analog modulation sequence with two

lev-els is used for an insensitive channel, that is, Sk = bT kak is

a circular matrix Examples of this type of channel include

the first two channels of an HM system Assuming that the

sequence errorsε k depend only on a k, that is, ε k,n1 = ε k,n2

whena(k,n1 ) = a(k,n2 ), the modulation vector can be written



ak = α kak+[β k β k · · · β k] for some values of the scaling

fac-torα kand offset term βk The channel matrixPk(z) for the

channel modulated with the sequenceakthen becomes



Pk(z) =bT k

α kak+

β k β k · · · β k

· Hk(z)

= α kSk· Hk(z) + β kBkHk(z), (19)

where Bkis a diagonal matrix consisting of the elements of

bk The first term is pseudocirculant, and thus the system is

insensitive to modulation sequence scaling factors in channel

k The impact of the o ffset term β k, that is, the second term,

is explained underSection 3.2.4below

3.2.2 Bilevel sequence for sensitive channels

Consider one of the subsetsC iin the partition of the channel

set The sum of the Sk-matrices corresponding to the

chan-nels in the set,

k ∈ C iSk, is a circulant matrix, whereas the constituent matrices are not Examples of this type of

chan-nels are the TIM systems and the HM systems with more than

two channels As in the insensitive case, the modulation

vec-tors are writtenak = α kak+ [β k β k · · · β k], and the sum of

the channel matrices for the channel subset becomes



k ∈ C i



Pk(z) = 

k ∈ C i

bT k

α kak+

β k β k · · · β k 

· Hk(z)

=





H0,i(z) ·

k ∈ C i

α kSk



k ∈ C i

β kBkHk(z),

(20)

where Bkis a diagonal matrix consisting of the elements of

bk The first sum is generally not a pseudocirculant matrix,

and the channels are thus sensitive to sequence gain errors If

the gains are matched, denoteα0,i = α k = α lwhenk, l ∈ C i,

the channel matrix sum may be written



k ∈ C



Pk(z) =



α0,iH0,i(z) ·

k ∈ C

Sk



k ∈ C

β kBkHk(z), (21)

x(n)

ε k(n)

×

+

a k(n)

γ k δ k

H k(z)

ΔH k(z)

×

+

b k(n)

y k(n)

Figure 6: Channel model with nonideal analog circuits

z −1

.

x(n) β k

↓ M

↓ M

Hk(z)

↑ M

↑ M

z z −1

+

.

.

↓ M

↓ M

Bk

↑ M

↑ M z

+

y k(n)

.

Figure 7: Model of errors in a parallel system pertaining to se-quence offsets

and it is seen that the first term is a pseudocirculant matrix, and the channel set is alias-free Again, the impact of the o ff-set termβ kis explained underSection 3.2.4below

3.2.3 Multilevel sequences

If an insensitive channel is modulated with a multilevel se-quenceak =ak+ε k, the channel matrix becomes



Pk(z) =bT k

ak+ε k



· Hk(z)

=Sk· Hk(z) + b T

k ε k· Hk(z),

(22)

which is pseudocirculant only if bT k ε kis a circulant matrix Systems with multilevel analog modulation sequences are thus sensitive to level errors

3.2.4 Modulation sequence offset errors

Consider here the modulation sequence offset errors intro-duced above under Sections3.2.1and3.2.2 The channel ma-trix for a channel with a modulation sequence containing an

offset error can be written as (19) Thus the error pertaining

to the sequence offset is additive, and can be modelled as in Figure 7 The signal is thus first filtered throughH k(z) and

then aliased by the system Bk, as Bk is not pseudocirculant

unless the elements in the digital modulation sequence bkare identical However, as the signal is first filtered, only signal components in the passband ofH k(z) will cause aliasing If

the signal contains no information in this band, aliasing will

be completely suppressed Typically the signal has a guard band either at the low-frequency or high-frequency region to allow transition bands of the filters, and the modulator can then be suitably chosen as either a lowpass type or highpass type, respectively Errors pertaining to sequence offsets are demonstrated inExample 1inSection 4

0 0.2π 0.4π 0.6π 0.8π π

ωT

−150

−100

−50

0

(a) Simulation using ideal system

ωT

−150

−100

−50

0

(b) Simulation with 2% gain mismatch in one channel

ωT

−150

−100

−50

0

(c) Simulation with 1% o ffset error in one modulation sequence

ωT

−150

−100

−50 0

(d) Simulation with 1% o ffset error in one modulation sequence using highpass modulators instead of lowpass modulators

Figure 8: Sensitivity of TIM ADC inExample 1

Channel offsets must be removed for each channel in order

not to overload theΣΔ-modulator Offsets affect the system

in a nonlinear way and may not be analyzed using the

multi-rate formulation However, the problem has been well

inves-tigated and numerous solutions exist [12,16,20]

In this section, examples of how the formulation can be used

to analyze a system’s sensitivity to channel mismatch errors

are presented Examples are provided for the TIM, HM, and

FBDM ADCs Also, an example is provided of how the

for-mulation can be used to derive a new architecture that is

in-sensitive to channel matching errors

Example 1 (TIM ADC) Consider a TIM ADC [3,4] with

four channels The samples are interleaved between the

channels, each encompassing identical second-order lowpass

modulators and decimation filters Ideally, their z-domain

transforms may be written

H k(z) = H(z) =

⎧

⎨

⎩

z −1, − π

4 ≤ ωT ≤ π

4,

All modulators are running at the input sampling rate, with

their inputs grounded between consecutive samples Thus

the modulation sequences are

a0(n) = b0(n) =1, 0, 0, 0, ,

a1(n) = b1(n) =0, 1, 0, 0, ,

a2(n) = b2(n) =0, 0, 1, 0, ,

a (n) = b (n) =0, 0, 0, 1, ,

(24)

all periodic with periodM =4 The vectors akand bkare as defined by (13):

a0=b3=1 0 0 0

a1=b0=0 0 0 1

a2=b1=0 0 1 0

a3=b2=0 1 0 0

(25)

The matrices Sk, defined by (12), then become

S0=bT0a0=

⎡

⎢

⎢

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

⎤

⎥

⎥,

S1=bT1a1=

⎡

⎢

⎢

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

⎤

⎥

⎥,

S2=bT

2a2=

⎡

⎢

⎢

0 0 0 0

0 0 1 0

0 0 0 0

0 0 0 0

⎤

⎥

⎥,

S3=bT

3a3=

⎡

⎢

⎢

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⎤

⎥

⎥.

(26)

Because the sum of all Sk-matrices is a circulant ma-trix, the system is alias-free and the transfer function for the system is given by (15) as H A(z) = z −1s0,13 H3,1(z4) = z −1

whereH3,1(z) =1 is the second polyphase component in the

polyphase decomposition ofH(z) The transfer function is

thus a simple delay, and the system will digitize the complete

spectrum

As none of the Sk-matrices are circulant, and a

circu-lant matrix can be formed only by summing all the matrices,

the TIM ADC requires matching of all channels in order to

eliminate aliasing Thus we defineC0 = {0, 1, 2, 3},

accord-ing to the description inSection 2.1.1 The system has been

simulated with modulator nonidealities and errors of bilevel

sequences for sensitive channels, as described in Section 3

Figure 8(a)shows the output spectrum for the ideal case with

no mismatches between channels (γ k =1 for allk)

Apply-ing 2% gain mismatch for one of the channels (γ0 = 0.98,

γ1= γ2= γ3=1), the spectrum inFigure 8(b)results, where

the aliasing components can be clearly seen InFigure 8(c),

the channel gains are set to one, and a 1% offset error has

been added to the first modulation sequence (β0 = 0.01,

β1= β2 = β3 =0), which results in aliasing InFigure 8(d),

high-pass modulators have been used instead, and the

distor-tions disappear, as predicted by the analysis inSection 3.2.4

Example 2 (HM ADC) Consider a nonoversampling HM

ADC [6] with eight channels In this case, every channel

fil-ter is an 8th-band filfil-ter (H k(z) = H(z), k =0, , 7) and the

modulation sequencesa k(n) and b k(n) are

a0(n) = b0(n) =1, 1, 1, 1, 1, 1, 1, 1, ,

a1(n) = b1(n) =1,−1, 1,−1, 1,−1, 1,−1, ,

a2(n) = b2(n) =1, 1,−1,−1, 1, 1,−1,−1, ,

a3(n) = b3(n) =1,−1,−1, 1, 1,−1,−1, 1, ,

a4(n) = b4(n) =1, 1, 1, 1,−1,−1,−1,−1, ,

a5(n) = b5(n) =1,−1, 1,−1,−1, 1,−1, 1, ,

a6(n) = b6(n) =1, 1,−1,−1,−1,−1, 1, 1, ,

a7(n) = b7(n) =1,−1,−1, 1,−1, 1, 1,−1, .

(27)

The vectors akand bkbecome

a0=b0=1 1 1 1 1 1 1 1

a1= −b1=1 −1 1 −1 1 −1 1 −1

a2=b3=1 −1 −1 1 1 −1 −1 1

a3= −b2=1 1 −1 −1 1 1 −1 −1

a4=1 −1 −1 −1 −1 1 1 1

b4=−1 −1 −1 −1 1 1 1 1

a5=1 1 −1 1 −1 −1 1 −1

b5=1 −1 1 −1 −1 1 −1 1

a6=1 1 1 −1 −1 −1 −1 1

b6=1 1 −1 −1 −1 −1 1 1

a7=1 −1 1 1 −1 1 −1 −1

b =−1 1 1 −1 1 −1 −1 1

(28)

With Sk =bT kak, the following matrices can be computed:

S0=1,

S1=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

−1 1 −1 1 −1 1 −1 1

−1 1 −1 1 −1 1 −1 1

−1 1 −1 1 −1 1 −1 1

−1 1 −1 1 −1 1 −1 1

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦ ,

S2+ S3=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦ ,

S4+ S5+ S6+ S7=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

(29)

It is seen that S0and S1are circulant matrices Also, S2+S3

is circulant Further, the remaining matrices sum to a

circu-lant matrix S4+ S5+ S6+ S7, whereas no smaller subset does Thus, in order to eliminate aliasing, the channels are parti-tioned into the setsC0 = {0},C1 = {1},C2 = {2, 3}, and

C3 = {4, 5, 6, 7} The HM ADC thus contains both insensi-tive channels 0 and 1, and sensiinsensi-tive channels 2, , 7.

Using the model of the ideal system, the spectrum of the output signal is as shown inFigure 9(a).Figure 9(b)shows the output spectrum for the system with 1% random gain mismatch (γ k ∈[0.99, 1.01]), where the aliasing distortions

are readily seen Matching the gains of the C2-channels to each other (settingγ2= γ3) and the gains of theC3-channels

to each other (settingγ4 = γ5 = γ6 = γ7), the spectrum in Figure 9(c)results, and the distortions disappear Although the HM ADC is less sensitive than the TIM ADC, the match-ing requirements for eight-channel systems and above are still severe Another limitation is that the reduced sensitiv-ity seemingly requires a number of channels that are a power

of two For Hadamard matrices of other orders, extensive searches by the authors have not yielded solutions with sim-plified matching requirements

Example 3 (FBDM ADC) For the FBDM ADC, the input

signal is applied unmodulated toN modulators converting

different frequency bands Consider as an example a four-channel system consisting of a lowpass four-channel, a highpass

0 0.2π 0.4π 0.6π 0.8π π

ωT

−150

−100

−50

0

(a) Simulation using ideal model

ωT

−150

−100

−50

0

(b) Simulation using 1% channel gain mismatch

ωT

−150

−100

−50

0

(c) Simulation using gain matching of sensitive channels

Figure 9: Sensitivity of TIM ADC inExample 2

ωT

−200

−150

−100

−50

0

50

Figure 10: Sensitivity of new scheme inExample 4 Simulation

us-ing 10% channel gain mismatch

channel, and two bandpass channels centered at 3π/8 and

5π/8.

As the signal is not modulated,

(31) for allk, and

Sk =

⎡

⎢

⎢

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

⎤

⎥

for allk As each S k-matrix is circulant, the system is insen-sitive to channel mismatches Further, modulation sequence errors are irrelevant in this case, as the signal is not modu-lated The FBDM ADC is thus highly resistant to mismatches Its obvious drawback, however, is the need to use bandpass modulators which are more expensive in hardware

Example 4 (generation of new scheme) This example

dem-onstrates that the formulation can also be used to devise new schemes, although a general method is not presented

A three-channel parallel system using lowpass modulators is designed The signal is assumed to be in the frequency band

− π/4 < ωT < π/4, and the ADC is thus an oversampled

sys-tem and is described according toSection 2.1.2withL =4 andM =8

Using complex modulation sequences, three bands of widthπ/4 centered at − π/4, 0, and π/4 can be translated to

baseband and converted with a lowpass ADC These modu-lation sequences area0(n) =1,a1(n) =exp(jπn/4), a2(n) =

exp(− jπn/4), and b k(n) = a ∗ k(n) Summing the resultant S k -matrices yields



Sk

=1+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

√

−2 − √2 0 √

− √2 −2 − √2 0 √

2

√

2

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

(33)

Unfortunately, using complex modulation sequences is not practical However, as the modulators and filters are identi-cal for all channels (H k(z) = H(z) for all k), any other choice

of modulation sequences resulting in the same matrix will perform the same function Moreover, for a decimated sys-tem, relaxations may be allowed on the new modulation se-quences In this case, with decimation by four, it is sufficient

to find replacing modulation sequences a k and b k such

that the sum of the resulting S k-matrices equals

Skon rows

4 and 8, as gcd(L, M) =4 One such choice of modulation se-quences is

a0=1 1 1 1 1 1 1 1

,

a1=1 1 0 −1 −1 −1 0 1

,

a2=1 0 0 0 −1 0 0 0

,

b0=0 0 0 1 0 0 0 1

,

(34)

b1=0 0 0 − √2 0 0 0 √

2

b2=0 0 0 (√

2−2) 0 0 0 (2− √2)

The analog modulation sequences a k can easily be im-plemented by switching or grounding the inputs to the

modulators, whereas the nontrivial multiplications in b k

can be implemented with high precision digitally Note that



b k T a k

=1 +

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

−2 − √2 0 √

2 0 − √2

2 0 − √2 −2 − √2 0 √

2

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

which is equal to

Skin (33) on rows 4 and 8 Note also that

the S k-matrices, given on rows 4 and 8 by



b0,3

b0,7



a0=



1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

 ,



b1,3

b1,7



a1=



− √2 − √2 0 √

2 0 − √2

√

2 0 − √2 − √2 − √2 0 √

2

 ,



b2,3

b2,7



a2=



(√

2−2) 0 0 0 (2− √2) 0 0 0 (2− √2) 0 0 0 (√

2−2) 0 0 0

 , (38) are circulant on these rows, and thus the system is insensitive

to channel mismatches This is demonstrated in Figure 10,

where the channel gain mismatch is 10% and no aliasing

re-sults However, as three levels are used in the analog

modu-lation sequences a1 and a2, the system is sensitive to

mis-matches in the modulation sequences of these channels, as

described inSection 3

The primary purpose of this paper is to investigate the signal

transfer characteristics of the parallelΣΔ-system However,

the system’s noise properties are also affected by the choice of

modulation sequences, and therefore a simple noise analysis

is included

A noise model of the parallelΣΔ-system can be depicted

as inFigure 11 The quantization noise q k(n) of channel k

is filtered through the noise transfer function NTFk(z) and

filterG k(z) The filtered noise is then modulated by the

se-quenceb k(n) The channels are summed to form the output

y(n).

In order to determine the statistical properties of the

out-puty(n), channel k is modeled as inFigure 12 Denoting the

spectral density of the quantization noise of channel k by

R Q k(e jω), the spectral densities of the polyphase components

y k,mof the channel output can be written

R y k,m



e jω

= b k,m2

M−1

l =0

G k,l

e jω2

R Q k



e jω

whereG k,l(z) are the polyphase components of the cascaded

system NTF (z)G (z) It is seen that the noise power is scaled

q0 (n)

q1 (n)

q N −1(n)

NTF0(z)

NTF 1 (z)

NTFN−1(z)

.

G0 (z)

G1 (z)

G N−1(z)

.

×

×

×

+

b0 (n)

b1 (n)

b N−1(n)

y(n)

Figure 11: Noise model of parallel system

q k(n)

NTFk(z) G k(z)

↓ M

↓ M

↓ M

↑ M

↑ M

↑ M

z −1

z −1

y k,0(m)



y k,1(m)



y k,M−1(m)



b k,0

b k,1

b k,M−1

+

.

.

+

z

z

y k(n)

Figure 12: Noise model of channk.

by the factorb k,m2 , and it is thus of interest to keep the ampli-tudes of the modulation sequences low on the digital side For example, inExample 4, alternative choices of a1and b2

would have been

a1=[0 1 0 −1 0 −1 0 1],

However, in this case the noise power is larger This shows that the smaller magnitudes of the digital modulation se-quences, as in (36), is preferable from a noise perspective

In this paper, a new general formulation of analog-to-digital converters using parallel ΣΔ-modulators was introduced The TIM-, HM-, and FBDM ADCs have been described

as special cases of this formulation, and it was shown how the model can be used to analyze the sensitivity to channel matching errors for a parallel system Both Nyquist-rate and oversampled systems have been considered, and it was shown that an oversampled system may have a reduced sensitivity

to mismatches, which may be determined using the formu-lation The usefulness of the formulation is not limited to analysis of existing schemes, but also for the derivation of new ones, which was exemplified